### abstract

- arency g(x,, y,) is placed at the first focal plane of a lens of infinite diameter (see Fig. 1). Using the Fresnel approximation, we obtain the three-dimensional complex ampli- tude distribution past the lens7: exp(jkz) u(x,y,z)- jAz f f g'(x,,y2) jk (1) where (x2, y, are the coordinates of the lens's plane and g'(x2, Y2) is the field amplitude just in front of the lens, which is given by (2) Substituting Eq. (2) into Eq. (1), changing the order of integration, and performing the integration over x2 and y2, we obtain u(x,y,z)= exp[jk(z + f)] /_ j Af f g(x, y) - . k(z - k yyl dxidy x exPb-j 2f2f)(x + y, 2) - j-(xx + . (3) We note that, at the second focal plane z = f, Eq. (3) yields the familiar Fourier integral relation between u(x,y,z = f) and g(x,y). Here we are interested in the field distribution along the z axis (x, y = 0). Changing to polar coordinates, (x, Yl) (r, 0), we see that Eq. (3) becomes u(z) = exp[jk(z + f] exp[ k(f).]d. 2jAf fo t(p) -j - = exp[j